Asymmetry of convex polytopes and vertex index of symmetric convex bodies
In [Geom. Dedicata 90, 45--48 (2002; Zbl 0999.52008)] the authors have shown that a~polytope in~\({\mathbb R}^n\) with few vertices is far from being symmetric with respect to the Banach--Mazur distance. More precisely, it was established that Banach--Mazur distance between such a~polytope and any symmetric convex body is large. In this paper they introduce a~new, averaging-type parameter, to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, and in fact it satisfies the same lower bound as the Banach--Mazur distance. In a~sense it exhibits the following phenomenon: if a convex polytope with small number of vertices is as close to a~symmetric body as it can be, then most of its vertices are as~bad as the worst one. The authors apply their results to provide a lower estimate on the vertex index of a~symmetric convex body, which was recently introduced in [Adv. Math. 215, No. 2, 626--641 (2007; Zbl 1131.46011)] by \textit{K.~Bezdek} and \textit{A.~E.~Litvak}. Furthermore, the affirmative answer is given to a~conjecture by \textit{K.~Bezdek} [Problem 6.3 of Period. Math. Hung. 53, 59--69 (2006; Zbl 1127.52026)] on the quantitative illumination problem.
scientific article; zbMATH DE number 5532462
scientific article